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Inverse Synthetic Aperture Radar (ISAR) image gating for RCS extraction using backprojection is compared with image gating using smoothed reweighted L1-optimization in this study. The RCS of an object is measured by placing the object placed on a turntable which is rotated in an angular range while sweeping the frequency in the desired frequency range. A common model with isotropic point scatterers fixed in the object coordinate system is used in the ISAR imaging process. This model is used to define a forward operator. The ISAR image can be formed by operating with the backpropagation operator (i.e. backprojection), the adjoint of the forward operator, on the measured RCS. This robust method to solve the inverse problem gives an image with a resolution limited by the frequency bandwidth and the angular range. The RCS for a scattering feature is commonly determined by using the forward operator on the point scatterers in the image that are determined to belong to the scattering feature in ISAR image gating. L1-optimization is a method that can be used to get images with higher resolution and hence better separation of the different scattering features than backprojection. L1-optimization is well suited for naturally sparse ISAR images. One method to mitigate that the scatterers are restricted to a fixed grid is to use smoothed reweighting [1]. L1-optimizations are performed consecutively in a few steps where a smoothed version of the previous solution is used to determine a weighting matrix for the next step. Smoothed reweighted L1-optimization gives images with better separation of the scattering features in the ISAR image. Simulated and measured RCS data are used to compare image gating using backprojection with gating using smoothed reweighted L1-optimization. The main conclusion of this study is that the RCS can be extracted for scattering features, not resolved in backprojection images, using the smoothed reweighted L1-optimization. [1] D. Pinchera and M. D. Migliore, “Accurate reconstruction of the radiation of sparse sources from a small set of near-field measurements by means of a smooth-weighted norm for cluster-sparsity problems,” Electronics, vol. 10, no. 22, p. 2854, 2021.
The RCS of a target can be estimated using electromagnetic modeling if accurate geometries and material descriptions are available. An exact numerical calculation often requires prohibitive processing times. Moreover, numerical predictions with approximate techniques are difficult as it is challenging to consider all the physical phenomena. Therefore, a suitable RCS measurement facility adapted to the target size and specifications is required to estimate the RCS of a given target and to validate the numerical predictions. In general, the measurement of RCS takes place in anechoic chambers that simulate free-space and far-field conditions and where the unwanted reflections (walls, target mount, objects in the range, and the target interactions) are reduced. This paper presents a broadband measurement and validation of the RCS of a metallic trihedral corner reflector of 30 cm sides when fully anechoic conditions are not available, and consequently, some undesirable echoes are present in the measurements. Firstly, the measurement facility calibration and the target calibration are outlined. A single target reference approach is performed using a sphere as a reference, and its scattering response is shortly described. Then, the measurement of the target is performed. After these steps, a processing procedure is applied to isolate the target response from the background and the close responses due to unwanted reflections. The post-processing technique and the acquisition system are presented and discussed. The measurements are performed at X band as a function of the viewing angle for vertical transmit and receive polarization. To validate the technique, the RCS of the trihedral corner reflector is numerically simulated using the Integral Solver (I-Solver) of CST, with a Gaussian excitation, for vertical transmit and receive polarization. Measurements are compared with results obtained from CST software and show a good agreement with the numerical simulations. This setup will be used for RCS measurement of different complex targets and compared with measurements from other facilities to analyze and evaluate the RCS measurement uncertainty.
Plane wave generator (PWG) for Over The Air (OTA) characterization of beamforming millimeter wave devices, provides an attractive solution comparing to conventional measurement techniques (Compact Antenna test Ranges (CATR) and Far-field chambers). MVG’s Plane wave generator for 5G NR FR2 applications ([1]-[4]) is an innovative tool which permits the user to measure the radiating elements with low to medium directivity radiation characteristics with excellent precision. Conventional CATR systems are not suited for stationary DUT (with / without person) measurement scenario. In this paper, experimental results are presented for a dual-polarized PWG system, covering the 3GPP bands n257, n258 and n261 (24.25-29.5 GHz). System measurement results show good comparison with simulations and measurements of the PWG alone. Another advantage of PWG presented here, is that we can modify the size of the QZ. Results from a pre-production unit for a 15cm QZ shows amplitude variation of less than ±1 dB and achieve more precision for smaller DUT. Measurement results from the pre-production unit with a quiet zone of up to 38cm sphere diameter, show amplitude variations of less than ±2dB. This variation is compatible with the DUT + phantom or human measurement application. Pattern results for Antenna Under Test (AUT) with low to medium directivity (6dBi up to 17dBi) compare well with simulations and measurements from other systems. For a given AUT, the impact of different positioning mast is also evaluated. Excellent stability of patterns, when the AUT is placed at different positions inside the QZ, is observed. These results confirm that the dual-polarized PWG system presents an attractive solution for FR2 characterization of low to medium directivity radiating elements.
The mission of the Naval Surface Warfare Center, Indian Head, Maryland, EOD Department, is to utilize the latest available technology in the advancement of Explosive Ordnance Disposal (EOD) equipment and techniques. This mission includes the test and evaluation of current and developmental systems, which will be discussed in this paper. EOD exploits multiple physical phenomena in its task of ordnance detection, including chemical and electromagnetic. Electromagnetics include RF fields, light (including laser, infrared and ultraviolet), and nuclear radiation. For each phenomena, there may be several different technologies used to provide multi-mode detection capability. This study focuses on the electromagnetic subset of detection RADAR, and specifically Ground Penetrating Radar (GPR), which is distinguished by its earth surface domain and generally downward field of view. The paper will give a very brief overview of GPR theory and equipment, its use in EOD, and then will focus on the RF test and measurement of electromagnetic fields generated by GPR systems and antennas. An RF antenna/system test plan will be detailed, along with the design and development of antenna gain and radiation pattern measurement techniques. The measured data from GPR technology will be graphically displayed, analyzed and compared in terms of the potential for GPR effectiveness.
In this paper, a 77 GHz microstrip comb-line antenna array for an automotive RADAR application with a low sidelobe level is proposed. The microstrip technology is used for the antenna due to its low fabrication cost, small size, and easy integration with other microwave circuitry. At very high frequencies such as millimeter waves, the gain of a single element patch antenna is not enough to withstand the RADAR application requirements, hence an array of antennae is beneficial. A The Phased array antenna configuration is needed to have a high gain and low sidelobe level of -20 dB and a beam steering mechanism. The design procedure used here is the implementation of a single comb antenna, that is further realized into a 1 x 10 uniform linear array of a comb line array. It has a gain of 14.81 dB and a sidelobe level of -15 dB. The radiation in the comb antenna is primarily due to the open sides with the lengths of the comb serving as transmission lines. The adjacent combs are placed at the distance of λ in order to co-phase the antenna elements at the desired frequency. Additionally, with an aim of reducing the sidelobe level, Taylor amplitude distribution is used, and the tapered array is designed. This methodology helped to achieve a sidelobe level of -20 dB. The gain of an overall array is increased to 20 dB by realizing the array of 4 x 10. Another requirement of the Automotive Radar is beam steering to accurately detect the target. Butler matrix is a beamforming network chosen to feed the phased array antenna. The proposed antenna array is simulated in Ansys HFSS with Rogers RO 3003 substrate of the thickness of 1.27 mm and has an overall dimension of 9 x 14.96 mm2. The goal of the design of this antenna is to acquire an appropriate radiation pattern with a low side lobe level better than -20dB and achieve beam steering using the Butler matrix to have a phased array configuration. Index Terms— RADAR, Antenna array, Comb line array, Butler Matrix, Phased array.
The precise characterization of the complex permittivity, particularly loss tangent, in low-loss dielectric samples at microwave frequencies usually employs resonant cavity methods, where the quality factor of some resonance is determined by a precisely dimensioned sample of the material placed inside the cavity. In order to characterize materials over a broad set of frequencies, a separate measurement fixture and sample is required for each frequency, a tedious and expensive endeavor. In response, one may turn to a single broadband measurement system, such as the focused beam system, but simple transmission and reflection measurements suffer from poor loss tangent sensitivity. In this work, a hybrid approach is investigated whereby a highly resonant periodic metallic array adjoined to a dielectric sample is measured in a broadband focused beam system. A frequency selective surface (FSS) is designed to be placed against a planar dielectric sample to create a transmission or reflection response that is sensitive to the loss tangent of the material under test. This sandwiched structure is illuminated by the focused beam system to approximate plane-wave-like incidence, and scattering parameters measured. It is shown that the magnitude of response at the resonant frequency is linearly dependent on the loss tangent of the material under test for a certain range of loss tangents, and sources of error that limit sensitivity at lower loss tangents are explored. The effect of various FSS design parameters on loss tangent sensitivity is investigated, including sample thickness, FSS substrate thickness and complex permittivity, and FSS element pattern. Techniques for extracting complex permittivity from the scattering parameters of the focused beam measurement are presented, along with measured permittivity data from the FSS against a variety of well-known materials.
The Boeing 9-77 Compact Radar Range has utilized low-frequency solutions since the 1990s. However, compact radar ranges have innate challenges when it comes to low-frequency measurements, typically due to facility size limitations. Due to increasing demand for more reliable data across a broad set of frequencies, an upgrade to the existing Ultra High Frequency (UHF) antenna feeds was designed and implemented in July 2020. This antenna was developed with field quality improvements, reliability, repeatability, and maintainability in mind. Unlike the previous design, this antenna was designed as an array with weighted feeds to complement the characteristics of the pre-existing range Gregorian reflector system. This new UHF antenna array leveraged the Weighted Element Method (WEM) along with extensive electromagnetic modeling and trade studies to achieve an efficient design at a minimum size. As a result of these design choices, the new antenna has doubled the efficiency in the band of interest. In addition, the frequency bandwidth of the antenna was improved while also reducing calibration and background drift. Lastly, this array has significantly improved the field quality of the quiet zone compared to the previous antenna system and improved the signal-to-noise ratio. This paper describes the UHF Antenna Array design process and the compact range measurements results to demonstrate the benefit of the WEM for feed arrays in a compact range. Additionally, the authors present an evaluation of methods used to create a digital twin of the UHF Antenna Array and a summary of best practices for future development of weighted antenna arrays for compact radar ranges.
One benefit of cooperative automated and connected driving lies in the fusion of multiple mobile wireless sensor and data transmission nodes, covering complementary technologies like radar, cellular and ad-hoc communications, and alike. Current developments indicate enormous potential to increase the environmental awareness through joint communication and radar sensing. In this respect, future channel models require knowledge of bi-static reflectivities of road users over a range of illumination and observation angles, both in the nearfield and in the far-field. To establish reference data and model such angle-dependent RCS variations, this paper deals with realistic pedal and wheel rotations of a bicycle based on electromagnetic simulations. In the simulation setup, idealized far-field conditions with plane-wave illumination and observation were assumed, while the angles covered the entire azimuth with 201 variations of the pedal and wheel positions. The fluctuation of the RCS is analyzed and discussed in terms of its probability density and cumulative distribution functions. Depending on the angular constellation, the range of the fluctuation varied between 1 dB and 14 dB, while the specular reflection and forward-scattering showed almost no fluctuation.
The radiation and scattering pattern characteristics of open-ended rectangular waveguide with a chamfered tip are examined. Despite common and widespread use as a probe antenna for planar near-field antenna measurements, a methodical investigation of the chamfered-tip design and resultant performance has not been published. A computational electromagnetics (CEM) model for an open-ended rectangular waveguide probe with a parameterized chamfered tip has been constructed and results for both radiation and scattering patterns are presented. A comparison of results includes a probe without a chamfer and a probe typical of that available from commercial suppliers. It is shown that, for a series of standard waveguide size probes sharing a common thickness for the waveguide wall and chamfered tip, the radiation pattern is relatively insensitive to the chamfer tip designs studied until frequency increases into W-band (WR-10). The scattering pattern characteristics for the same series of standard waveguide size probes show a reduction in on-axis (boresight) monostatic radar cross section (RCS) for chamfered tip waveguides compared to blunt-ended waveguides and that this reduction increases for increasing frequency.
We present a fast source reconstruction method suitable for antenna diagnostic applications of radiating structures on electrically large platforms. The method is based on a novel implementation of a recent reformulation of the inverse electromagnetic scattering problem, and is solved using a Higher Order Method of Moments (MoM) discretization. The novel implementation achieves asymptotically better scaling the previously possible, and in particular the memory use is substantially lower than was previously possible. Results from two example cases are presented where the new method is compared to the current commercial state-of-the-art solver in DIATOOL 1.1, and significant improvements are observed in terms of computation times and memory requirements.
Instrumentation radar metrology waveform techniques that simultaneously transmit two orthogonal sequences of orthogonal electromagnetic polarizations are explored for applicability toward both static and dynamic RCS signature and ultra-wideband imaging measurements using simultaneous H-pol and V-pol (SHV) waveforms. Static, pulsed measurements with independent transmit polarizations are modulated and radiated; reflections from a depolarizing target are measured where the return signals are coherently combined. Each transmit polarization is independently modulated using a diverse phase sequence, which leaves a unique “fingerprint” by which the orthogonal polarization separation is achieved. Using only the coherent combination and associated transmit and receive RF channel characterizations, the original measurements are reconstructed. Simulations serve as a baseline for measured results, from generating pure SHV waveforms and then providing simultaneous full scattering matrix (FSM) measurements, in order to achieve greater purity of FSM signatures, while reducing measurement times by a factor of two.
A Spherical Passive/Active Radar Calibration System (SPARCS) has been designed as an advanced, airborne, radar calibration device (CD). SPARCS is currently under development as an autonomous, battery-powered, high-endurance, flying platform. This self-contained, multi-function radar calibration and diagnostic system functions as 1) a Passive Spherical Reflector, 2) an Active RF Repeater, 3) a Synthetic Target Generator, and 4) an UWB RF Sensor and Data Recorder of the radar under test or the localized RF environment. This innovative CD exploits major advances in commercial technology during the past decade associated with autonomous airborne drones and miniaturized digital RF systems on chips (RFSoCs), and other miniature electronics. Emphasis has been placed on a recoverable, reusable CD that enables precision calibrations over extensive open-air test volumes used for dynamic aircraft RCS measurement, test and verification, or time space position information (TSPI) test range tracking radars. This paper highlights early efforts to parameterize and develop SPARCS, including advances in autonomous navigation and flight time, electric ducted fan performance, radar screens for thruster inlet and outlet ports, active calibration functionality, and improving calibration uncertainty. SPARCS will provide an unprecedented capability for radar instrumentation calibration, target emulation, environmental assessment and in situ, real-time calibration.
A set of Dual-Polarized Antennas with a 3:1 operating bandwidth has been developed for use in near-field ranges as the probe or range antenna and for use as a Compact Antenna Test Range (CATR) feed [1]. Key development parameters of the antenna are: a wideband impedance match to the coaxial feed line, E and H-plane 1 dB beam widths in excess of 30 degrees, -30 dB on axis cross-polarization, minimum polarization tilt and a phase center that varies over a small region near the aperture. To accomplish these design parameters, a family of range antennas has been developed and previously introduced. Two versions of the antenna have been manufactured and tested for performance. A 2-6 GHz version has been developed using traditional machining techniques and a 6-18 GHz version has been produced using additive manufacturing (3D printing) techniques [4]. These antennas provide proper illumination of the quiet zone for compact ranges used for antenna measurements as well as radar cross section (RCS) measurements. For RCS measurements, an additional requirement for time-based energy storage performance is considered. Energy storage in the feed can result in a pulse spreading or additional copies of the pulse in time, resulting in poor performances of the target characterization. This effect is called ‘ringdown’. In this paper, we focus on the RCS ringdown performance of the 6-18 GHz antenna produced using additive manufacturing. The measured performance of the antenna will be presented and discussed. Finally, the applicability of the antenna as a CATR feed for RCS measurements will be discussed.
A set of Dual-Polarized Antennas with a 3:1 operating bandwidth has been developed for use in near-field ranges as the probe or range antenna and for use as a Compact Antenna Test Range (CATR) feed [1]. Key development parameters of the antenna are: a wideband impedance match to the coaxial feed line, E and H-plane 1 dB beam widths in excess of 30 degrees, -30 dB on axis cross-polarization, minimum polarization tilt and a phase center that varies over a small region near the aperture. To accomplish these design parameters, a family of range antennas has been developed and previously introduced. Two versions of the antenna have been manufactured and tested for performance. A 2-6 GHz version has been developed using traditional machining techniques and a 6-18 GHz version has been produced using additive manufacturing (3D printing) techniques [4]. These antennas provide proper illumination of the quiet zone for compact ranges used for antenna measurements as well as radar cross section (RCS) measurements. For RCS measurements, an additional requirement for time-based energy storage performance is considered. Energy storage in the feed can result in a pulse spreading or additional copies of the pulse in time, resulting in poor performances of the target characterization. This effect is called ‘ringdown’. In this paper, we focus on the RCS ringdown performance of the 6-18 GHz antenna produced using additive manufacturing. The measured performance of the antenna will be presented and discussed. Finally, the applicability of the antenna as a CATR feed for RCS measurements will be discussed.
The future of cooperative automated and connected driving lies in the fusion of multiple mobile wireless sensor and data transmission nodes, covering technologies like radar, cellular and ad-hoc communications, and alike. Current developments indicate enormous potential to increase the environmental awareness through joint communication and radar sensing. In this respect, future wireless channel models aim at including bi-static reflectivities of road users, depending on different illumination and observation angles, in the nearfield as well as in the far-field. The limitations of the measurement distances within anechoic chambers unavoidably induce nearfield effects, especially for electrically large radar objects like realistic road users, and conventional bi-static RCS calibration techniques would eventually fail. In order to model the transition from the nearfield to the far-field reflectivity of road users, this paper uses the object scaling approach, with combined measurements and electromagnetic simulations. Bi-static reflectivity measurements of selected vulnerable road users are described, from the chamber setup all the way up to data post-processing. The approach of electromagnetic object scaling is applied to such bi-static reflectivity measurements, and the results are evaluated and discussed in comparison with numerical simulations. Initial proof-of-concept measurements of differently sized metal spheres confirmed the applicability of the scaling approach under far-field conditions very convincingly. Based on this, scaled models of radar objects, namely a bicycle and a pedestrian, were 3D printed and then metallized with copper paint. Compared to corresponding electromagnetic simulations of the original bi-static reflectivity of the radar objects, the results measured for the scaled models show very promising agreement with the numerical expectation. This study contributes to the further development of future wireless channel models considering bi-static multipath components of different road users, being an indispensable prerequisite to enhance the safety in future road traffic.
Motivation and background: With the increasing abundance and functionality of wireless communication systems, the requirements for virtual electromagnetic environments like shielded anechoic chambers, and the complexity of the test procedures increase accordingly. The scattering behavior of microwave absorbers is an essential indicator of their quality and suitability for use in such anechoic chambers. Current research activities deal with the revision of the IEEE standard 1128 on recommended practice for absorber characterization and give room for improved test procedures. Objectives and methods: In this paper, the angle-dependent backscattering of microwave absorbers was studied experimentally with respect to their different geometric shapes and material parameters. The dielectric permittivity of pyramidal and flat absorbers was measured between 1 GHz and 10 GHz, followed by systematic monostatic reflectivity measurements. Signal post-processing, including phase-coherent background subtraction and time-domain gating, were applied to minimize unwanted reflections and extract the wanted scattered signals. The radar cross-section (RCS) method was applied to derive the reflectivity with respect to different illumination angles for parallel and perpendicular polarizations. The results were compared to supplier specifications, electromagnetic simulations of the reflectivity, and the scattering pattern of a metal plate. Results and conclusions: The measurement results agree well with the numerical simulations. The data reveal that the reflectivity patterns of microwave absorbers are governed by their geometric shape, while the material properties do not modify the angular dependences qualitatively but result in a quantitative offset. Our findings help to improve the accuracy of monostatic RCS and absorber reflectivity measurements even further and lead to a better understanding of the physical origin of the scattering phenomena of microwave absorbers in general. Future work will extend our studies towards bi-static angle-dependent reflectivity measurements, in order to establish a consistent and comprehensive method for characterizing different types of microwave absorbers with respect to type, frequency, angle of illumination, angle of observation, and polarization. This research has been funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under the grants HE3642/14-1 and BO4990/1-1 (Electromagnetic modeling of microwave absorbers - EMMA).
There have been a number of numerical analyses of RF absorber presented in the literature. These analyses, however, tend to focus on the reflectivity of the material and not on the radar cross section (RCS) that it presents. Brumley studied the RCS of RF absorbers for the purpose of estimating the background RCS of anechoic ranges [1]. The study was done empirically; obtaining measurements of the RF absorber and looking at the RCS of different pyramids and wedges, with and without paint. Brumley presents some potential methods to improving the RCS signature of the range, thus reducing the background RCS of the site. In this paper, the suggestions presented by Brumley are revisited. Specifically, his recommendation for the twisted pyramid configuration which he was unable to measure due to the lack of absorber samples available for use in the test. In addition to the twisted pyramid, Brumley's approach of inserting smaller pyramids in the valleys of a larger pyramidal arrangement to reduce the edges parallel to the incoming wave are also presented. Different carbon loadings are modeled for the inserted pyramids. One is the standard loading of the inserted pyramid, and the other is the same loading as the main larger pyramidal arrangement such that all the absorber on the wall has the same material properties. Numerical studies are performed using time domain techniques as well as frequency domain techniques. The model is validated by comparing the RCS of a flat square plate with the theoretical solution. The results validate the data and the suggestions presented in [1] and present ways of improving some of the solutions by adjusting the material properties of the absorber.
Financial impacts often drive decisions to repurpose existing ranges instead of procuring new measurement facilities. These existing ranges have fixed geometries (height, width and length) that were set when the range was originally constructed and often are designed for a different purpose. The inability to set the geometry precludes the range designer from using the range geometry to improve measurement performance. Thus, the performance of the range is mostly dependent on the RF absorber and the range antenna directivity. In rectangular-shaped ranges for example, the lateral surfaces, side walls, ceiling and floor, are the critical surfaces to address in RF absorber arrangement. In this paper, numerical analyses of Chebyshev arrangements as well as dragon tail or tilted absorber are studied. This paper also analyzes the performance of Chebyshev absorber for normal incidence and for oblique incidence along with the proper arrangement of the Chebyshev period. While certainly these have been discussed previously in the literature, this paper consolidates the previous information and illustrates it with numerical examples to help the reader understand the best approach to use when repurposing a range.
The Austin RCS Benchmark Suite has recently been introduced to enable quantitative and objective comparison of computational systems for solving electromagnetic scattering problems, particularly, those relevant to aerospace applications. In the last year, five sets of problems were added to it: dielectric almonds (problem set III-B), mixed material almonds (III-C, III-D), perfectly electrically conducting (PEC) aircraft models (IV-A), and dielectric aircraft models (IV-B). For each problem set, a range of lengths and frequencies of interests are identified, interesting features are highlighted, and datasets containing reference results (from measurements, analytical methods, or numerical methods) are shared online. Although data from several radar cross section (RCS) measurement campaigns of non-metallic targets are available in the literature, these lack the information necessary to precisely model the materials, target geometries, and measurement setups, to quantify uncertainties in the data, and to identify appropriate directions for improving computational methods' performance. This limits their utility for benchmarking computational systems. This article presents an expansion of the Suite to include problems with more complex materials and reference results from a measurement campaign that attempted to ameliorate the deficiencies of existing datasets. Specifically, a set of thin-plate problems are added to the Austin RCS Benchmark Suite to increase material diversity. These consist of problem sets II-B: thin perfect-electrically conducting (PEC) plates, II-C: thin dielectric plates, II-D: thin magnetic radar absorbing material (MagRAM) plates, and II-E: thin MagRAM-coated PEC plates. Reference RCS data that enables validation, RCS measurement and material property uncertainty quantification, and benchmarking are also provided by conducting a simulation-supported measurement campaign in a compact range. To facilitate reproducibility, a popular low-loss dielectric material and a commercially available MagRAM were chosen for these problems: The dielectric material for problem set II-C is PolymaxTM polylactic acid (PLA). For problem sets II-D/E, the ARC Technologies' DD-13490 material is used. Thin plates were manufactured and their RCS were measured at Lockheed Martin's Rye Canyon Facility. The monostatic RCS measurement results and supporting simulation results are available online. Performance data for simulations as well as RCS measurement results with accompanying uncertainty will be presented for problem sets II-B/C/D/E at the conference.
A near-field far-field transformation (NFFFT) technique with a plane-wave synthesis is presented for predicting two-dimensional (2D) radar cross sections (RCS) from multistatic near-field (NF) measurements. The NFFFT predicts the FF of the OUT illuminated by each single source, then the plane-wave synthesis predicts the FF of the OUT each illuminated by each plane-wave by synthesizing the FFs given in the NFFFT step. The both steps are performed in the similar computational procedure based on a single-cut NFFFT technique that has been proposed previously. The method is performed at low cost computation because the NF and source positions are required only on a single cut plane. The formulation and validation of the method is presented.